91Ó°ÊÓ

If \(T_{s} \approx T_{\text {sur }}\) in Equation \(1.9\), the radiation heat transfer coefficient may be approximated as $$ h_{r, a}=4 \varepsilon \sigma \bar{T}^{3} $$ where \(\bar{T} \equiv\left(T_{s}+T_{\text {sur }}\right) / 2\). We wish to assess the validity of this approximation by comparing values of \(h_{r}\) and \(h_{r, a}\) for the following conditions. In each case, represent your results graphically and comment on the validity of the approximation. (a) Consider a surface of either polished aluminum ( \(\varepsilon=\) \(0.05)\) or black paint \((\varepsilon=0.9)\), whose temperature may exceed that of the surroundings \(\left(T_{\text {sur }}=25^{\circ} \mathrm{C}\right)\) by 10 to \(100^{\circ} \mathrm{C}\). Also compare your results with values of the coefficient associated with free convection in air \(\left(T_{\infty}=T_{\text {sur }}\right)\), where \(h\left(\mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\right)=0.98 \Delta T^{1 / 3}\). (b) Consider initial conditions associated with placing a workpiece at \(T_{s}=25^{\circ} \mathrm{C}\) in a large furnace whose wall temperature may be varied over the range \(100 \leq\) \(T_{\text {sur }} \leq 1000^{\circ} \mathrm{C}\). According to the surface finish or coating, its emissivity may assume values of \(0.05\), \(0.2\), and \(0.9\). For each emissivity, plot the relative error, \(\left(h_{r}-h_{r, a}\right) / h_{r}\), as a function of the furnace temperature.

Step by step solution

01

Calculate the exact radiation heat transfer coefficient

Define the exact radiation heat transfer coefficient as: $$ h_{r}=\varepsilon \sigma\left(T_{s}^{4}-T_{\text {sur }}^{4}\right) /\left(T_{s}-T_{\text {sur }}\right) $$

02

Calculate the approximated radiation heat transfer coefficient

Define the approximated radiation heat transfer coefficient as: $$ h_{r, a}=4 \varepsilon \sigma \bar{T}^{3} $$ where \(\bar{T} \equiv\left(T_{s}+T_{\text {sur }}\right) / 2\).

03

Step 3:Calculate the convective heat transfer coefficient in air for (a) only

Define the convective heat transfer coefficient in air as: $$ h\left(\mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\right)=0.98 \Delta T^{1 / 3} $$

04

Calculate and plot the radiation and convection coefficients for condition (a)

To do this, select the values of emissivity, surface, and surrounding temperatures and iterate through the given range. Plot the values of \(h_{r}\) and \(h_{r,a}\), comparing them with the convection coefficients.

05

Calculate and plot the relative error for condition (b)

To do this, select the emissivities and surface and surrounding temperatures for each case, and iterate through the given range of surrounding temperatures. Calculate the relative error as \((h_{r} - h_{r, a})/h_{r}\). Plot this error as a function of the furnace temperature.

06

Step 6:

Analyzing the plots, comment on the validity of the approximation in each situation (conditions (a) and (b)). This will show whether the approximation is reasonable for the given parameters or not.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Heat Transfer

Heat transfer is the process of thermal energy moving from one place to another due to a temperature difference. There are three primary mechanisms of heat transfer: conduction, convection, and radiation.
Conduction is the transfer of heat between substances in direct contact. Convective heat transfer involves the movement of fluids or gases, carrying heat with them. Lastly, thermal radiation is the emission of electromagnetic waves, which carry energy away from the emitting surface.
Understanding these mechanisms is essential for solving problems related to temperature changes in different materials and environments. In the context of the exercise, the focus is on radiation and convection as the main forms of heat transfer occurring between a surface and its surroundings.

Thermal Radiation

Thermal radiation is heat transfer through electromagnetic waves emitted by a body due to its temperature. This form of heat transfer does not require any medium, as it can occur even in a vacuum.
The amount of energy radiated depends on the temperature of the surface, its area, and the material's emissivity. Emissivity is a measure of how effectively a surface emits thermal radiation compared to a perfect black body, which has an emissivity of 1.
In the exercise provided, the Stefan-Boltzmann law is used to relate the surface temperature and emissivity to the energy radiated. A higher surface temperature results in more radiation being emitted.

Emissivity

Emissivity is a dimensionless quantity that measures how effectively a surface emits thermal radiation relative to a perfect black body. It ranges from 0 to 1, with 1 representing a perfect emitter and absorber of radiation, such as a black body.
Materials with a low emissivity, like polished aluminum, emit less radiation, while materials with high emissivity, such as black paint, emit more radiation. This characteristic plays a significant role in the heat transfer calculations demonstrated in the exercise, affecting the radiation heat transfer coefficient, which in turn impacts the temperature change of objects.

Convective Heat Transfer

Convective heat transfer is the movement of heat through a fluid (which can be a liquid or a gas) caused by molecular motion. When a surface is heated unevenly, the fluid surrounding it will become buoyant when heated, which induces motion and mixes the fluid, distributing the heat.
The convective heat transfer coefficient is a measure of the convective heat transfer per unit area and temperature difference. It's influenced by properties of the fluid, the flow conditions, and the surface geometry. As seen in the exercise, for free convection in air, it's described by a relationship involving the temperature difference raised to the power of one-third, which derives from the natural convection equations.